Calculus Examples

$f (x) = y^{3} - 3 y^{2} - x^{2} - 9 y + 10 x - 60$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $y^{3} - 3 y^{2} - x^{2} - 9 y + 10 x - 60$ with respect to $y$ is $\frac{d}{d y} [y^{3}] + \frac{d}{d y} [- 3 y^{2}] + \frac{d}{d y} [- x^{2}] + \frac{d}{d y} [- 9 y] + \frac{d}{d y} [10 x] + \frac{d}{d y} [- 60]$ .

$\frac{d}{d y} [y^{3}] + \frac{d}{d y} [- 3 y^{2}] + \frac{d}{d y} [- x^{2}] + \frac{d}{d y} [- 9 y] + \frac{d}{d y} [10 x] + \frac{d}{d y} [- 60]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 3$ .

$3 y^{2} + \frac{d}{d y} [- 3 y^{2}] + \frac{d}{d y} [- x^{2}] + \frac{d}{d y} [- 9 y] + \frac{d}{d y} [10 x] + \frac{d}{d y} [- 60]$

Step 2

Evaluate $\frac{d}{d y} [- 3 y^{2}]$ .

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Step 2.1

Since $- 3$ is constant with respect to $y$ , the derivative of $- 3 y^{2}$ with respect to $y$ is $- 3 \frac{d}{d y} [y^{2}]$ .

$3 y^{2} - 3 \frac{d}{d y} [y^{2}] + \frac{d}{d y} [- x^{2}] + \frac{d}{d y} [- 9 y] + \frac{d}{d y} [10 x] + \frac{d}{d y} [- 60]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$3 y^{2} - 3 (2 y) + \frac{d}{d y} [- x^{2}] + \frac{d}{d y} [- 9 y] + \frac{d}{d y} [10 x] + \frac{d}{d y} [- 60]$

Step 2.3

Multiply $2$ by $- 3$ .

$3 y^{2} - 6 y + \frac{d}{d y} [- x^{2}] + \frac{d}{d y} [- 9 y] + \frac{d}{d y} [10 x] + \frac{d}{d y} [- 60]$

Step 3

Since $- x^{2}$ is constant with respect to $y$ , the derivative of $- x^{2}$ with respect to $y$ is $0$ .

$3 y^{2} - 6 y + 0 + \frac{d}{d y} [- 9 y] + \frac{d}{d y} [10 x] + \frac{d}{d y} [- 60]$

Step 4

Evaluate $\frac{d}{d y} [- 9 y]$ .

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Step 4.1

Since $- 9$ is constant with respect to $y$ , the derivative of $- 9 y$ with respect to $y$ is $- 9 \frac{d}{d y} [y]$ .

$3 y^{2} - 6 y + 0 - 9 \frac{d}{d y} [y] + \frac{d}{d y} [10 x] + \frac{d}{d y} [- 60]$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$3 y^{2} - 6 y + 0 - 9 \cdot 1 + \frac{d}{d y} [10 x] + \frac{d}{d y} [- 60]$

Step 4.3

Multiply $- 9$ by $1$ .

$3 y^{2} - 6 y + 0 - 9 + \frac{d}{d y} [10 x] + \frac{d}{d y} [- 60]$

Step 5

Differentiate using the Constant Rule.

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Step 5.1

Since $10 x$ is constant with respect to $y$ , the derivative of $10 x$ with respect to $y$ is $0$ .

$3 y^{2} - 6 y + 0 - 9 + 0 + \frac{d}{d y} [- 60]$

Step 5.2

Since $- 60$ is constant with respect to $y$ , the derivative of $- 60$ with respect to $y$ is $0$ .

$3 y^{2} - 6 y + 0 - 9 + 0 + 0$

Step 6

Combine terms.

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Step 6.1

Add $3 y^{2} - 6 y$ and $0$ .

$3 y^{2} - 6 y - 9 + 0 + 0$

Step 6.2

Add $3 y^{2} - 6 y - 9$ and $0$ .

$3 y^{2} - 6 y - 9 + 0$

Step 6.3

Add $3 y^{2} - 6 y - 9$ and $0$ .

$3 y^{2} - 6 y - 9$